Characterization of projective spaces and $\mathbb P^r$-bundles as ample divisors

TL;DR

This paper proves that a projective manifold with an ample subsheaf in its tangent bundle must be projective space, and classifies manifolds with a projective bundle as an ample divisor.

Contribution

It establishes a characterization of projective spaces via tangent bundle properties and classifies certain manifolds with ample divisors using recent results.

Findings

If $T_X$ contains an ample subsheaf, then $X$ is isomorphic to $P^n$.

Classifies manifolds with a $P^r$-bundle as an ample divisor.

Provides a link between tangent bundle positivity and the structure of the manifold.

Abstract

Let $X$ be a projective manifold of dimension $n$. Suppose that $T_X$ contains an ample subsheaf. We show that $X$ is isomorphic to $\mathbb{P}^n$. As an application, we derive the classification of projective manifolds containing a $\mathbb{P}^r$-bundle as an ample divisor by the recent work of D.~Litt.